Question

The generator matrix G of a (6,3) linear block code (LBC) is given as
$G={\left[\begin{array}{cccccc}1& 0& 0& 1& 1& 0\\ 0& 1& 0& 0& 1& 1\\ 0& 0& 1& 1& 0& 1\end{array}\right]}_{3\times 6}$
Which of the following statement(s) is are correct?

• A. The code efficiency of LBC is 50%
• B. The minimum distance, $d_{min}\le 4$
• C. The LBC can detect atleast 2 errors.
• D. The largest hamming weight among the code words is 5

Answer 1

Answer: (A), (B)

The minimum distance, $d_{min}\le 4$

$$\begin{array}{ccc}d& C=dG& \text{Weight}\\ 000& 000000& 0\\ 001& 001101& 3\\ 010& 010011& 3\\ 011& 011110& 4\\ 100& 100110& 3\\ 101& 101011& 3\\ 110& 110101& 4\\ 111& 111000& 3\end{array}$$
$\therefore$ The largest hamming weight is 4
$d_{min}=3$
$d_{min}\le t+1$
where t = number of error which can be detected
$(\therefore t\ge 2$
$(\therefore$ The LBC can detect upto 2 errors
$d_{min}\le n-k+1$
$d_{min}\le 3+1=4$
Code efficiency = $\frac{k}{n}=\frac{3}{6}=\frac{1}{2}$ {n =6, k =3}
$=50\%$